A fast algorithm for 3D azimuthally anisotropic velocity scan |

The right-hand side of equation 5 is a quotient of two (discrete) generalized Radon transforms (Beylkin, 1984). They can be expressed in a unified way as (to simplify the notation, we write , in this and next subsections)

where is or some composite function of .

To construct the fast algorithm, we first rewrite equation 7 in the frequency domain as

where is frequency and is the Fourier transform of in time. We next perform a linear transformation to map all discrete points in and domains to points in the unit cube ; i.e., a point minmaxminmaxminmax is mapped to via

maxminmin | |||

maxminmin | |||

maxminmin |

a point minmaxminmaxminmax is mapped to via

maxminmin | |||

maxminmin | |||

maxminmin |

If we define a phase function as

(9) |

then equation 8 can be recast as

(throughout the paper, and are used to denote either sets of discrete points or the cubic domains containing them).

A fast algorithm for 3D azimuthally anisotropic velocity scan |

2015-03-27